TRANSACTIONS OF THE SECTIONS. 43 



of light directed level so that if it -were to traverse a sti-aight path it would pass 

 along an infinitely thin lamina of uniform density, hut with less density above and 

 greater below, tcouhl he lent hy virtue of the difference of the densities above and 

 below it. 



It must, however, be admitted that there is somethiug perplexing, or not quite 

 satisfactoi-y to the mind, in taking this final step to the perfectly level ray ; for as 

 soon as the inclination of the ray becomes zero the whole foundation and frame- 

 work of the investigation fails, there being then no oblique passage of a ray from 

 one lamina into another, no incident and no refracted ray, and consequently no 

 ratio of sines of angles of incidence and refraction; though all these would be 

 required to be discussed as if they existed in the case of every ray whose curvature 

 is to be compared with that of any other. Still, as both Professor Purser and the 

 author thought at the time, the investigation made the physical conclusion as to 

 level I'ays seem highly probable ; since, i'f it proves, as it seems to do, that a ray of 

 light descending obliquely must move along a certain curved path, and tliat the 

 curvature must increase as the inclination approaches towards horizontality, and 

 also that the rate of change of curvature with cliange of inclination approaches 

 towards zero as the inclination approaches towards horizontality, it must follow that 

 a ray of light passing exactly level will be bent with the same ciuwature as one in- 

 finitely nearly level. 



Several years later (in February 1870) a new investigation occurred to the author 

 of the present paper. The new one is much simpler, and it is more general, and its 

 reasoning holds good alike for level as for inclined rays. In fact the previous 

 investigation, founded on the ratio of the sines of angles of incidence and refraction, 

 and therefore in principle having no direct applicability to level rays, comes, when 

 considered in connexion with the new one, to be a case of this more general one, 

 seeing that under the undulatorj'- theory of light the proportionality of the sines of 

 the angles of incidence and refraction is not an ultimate fact or principle, but a 

 consequence of retardation of the velocity of light in the denser medium. In the 

 new investigation which will now be submitted the retardation of the velocity of 

 light in the denser medium is taken as the basis of the reasoning. 



Let M N and P be two level surfaces in the atmosphere, and let each of these 

 be supposed to pass through air of uniform density throughout each of them. They 

 may be conceived to be at a very small distance apart, 

 and then obviously a ray in descending obliquely from Fig. 2. 



one to the other will alter its cuiwature only by a very 

 slight amount. 



The fimdamental assumptions on which the investi- 

 gation will be based are the following three : — 



(1) It is assumed that the light at A has a certain 

 velocity, which may be called i\, and that the light at B, 

 where the air is denser, has a smaller velocity, which 

 may be called v^, 



(2) It is assumed that these velocities are constant 

 for all inclinations of the ray of light ; or, in other words, 

 that the velocity of the ray of light is independent of the 

 inclination of the ray to the horizontal strata of the air. 



(3) It is assumed that the direction of tlie light is per- 

 pendicular to the wave front, or that a surface taken 

 crossing every ray in a pencil of rays perpendicularly, 

 and then conceived to advance along the course of each 

 ray with the velocitj' of that ray, will continue to cross 

 every ray perpendicularly. 



Now let A B and C D be two successive positions, indefinitely near to each otlier, 

 of the advancing front of a ray or pencil of light whose direction of advance is in- 

 dicated by the lines E A and F B, and hy the arrows II in the figiu'e, the direction 

 at all points of AB being normal to the plane represented hy AB. Let the incli- 

 nation of A B to the vertical line B H be denoted by 6, which will then also denote 

 the inclination of the ray to the horizon. Let the thickness of the lamina of air 

 from MN to P be denoted by X, or let BH in the figure be denoted by X. 



